Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近[英文]

本文研究了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近问题，构造了定常Navier-Stokes方程对称破坏分歧点扩充系统及其谱Galerkin逼近扩充系统，证明了谱Galerkin逼扩充系统解的存在性和收敛性，从而给出了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近，并给出了逼近的误差估计。     

二重高阶对称破缺分歧点和它们的数值确定

本文考虑带Z_××Z₂对称性的两参数非线性的二重高阶对称破缺分歧点。利用对称性,我们提出了相应的正则扩张系统来确定这类分歧点。同时指出存在两条平方音叉式分歧点的道路通过该点。     

计算对称破缺Takens—Bogdanov为的有效方法

我们将提出一种直接方法来计算对称破缺Ｔａｋｒｎｓ－Ｂｏｇｄａｎｏｖ分歧点，这处方法构造了不引进零向量作为变量的不扩张系统，从而减少了计算量节约了内存，数值例子计算成功地说明了方法的有效性。     

计算圆域上p-Henon方程边值问题多个正解的分歧方法

首先应用分歧方法给出计算p-Henon方程边值问题O(2)对称正解的算法,然后以p-Henon方程中的参数l为分歧参数,在O(2)对称正解解枝上用扩张系统方法求出对称破缺分歧点,进而用解枝转接方法计算出其它具有不同对称性质的正解．     

计算Henon方程多个正解的分歧方法

首先应用分歧方法给出计算Henon方程边值问题D₄对称正解的3种算法, 然后以Henon方程中的参数r为分歧参数, 在D₄对称正解解枝上用扩张系统方法求出对称破缺分歧点, 进而用解枝转接方法计算出其他具有不同对称性质的正解.     

计算立方体上Henon方程多个正解的分歧方法

本文首先应用分歧方法给出计算立方体上Henon方程边值问题D₄(3)对称正解的三种算法, 然后以Henon方程中的参数r为分歧参数, 在D₄(3)对称正解解枝上 用扩张系统方法求出对称破缺分歧点, 进而用解枝转接方法计算出其它具有不同对称性质的正解.     

计算对称破缺Takens-Bogdanov点的有效方法

我们将提出一种直接方法来计算对称破缺Takens-Bogdanov分歧点,这种方法构造了不引进零向量作为变量的小扩张系统,从而减少了计算量并节约了内存,数值例子的计算成功地说明了方法的有效性。     

一种计算二重X0—对称破缺分歧点的有效方法

本文将构造一种直接的方法来计算二重Ｘ０－对称破缺分歧点,其构造的扩张系统没有将零向量引入当成变量,因此节约了计算的工作量和计算机内存,数值例子的实际显示了该方法是有效的。     

N阶分歧问题的数值分析

构造了求解一类非退化分歧点及相关参数的扩充系统，给出了拟牛顿迭代法并证明了收敛性．     

Ⅳ阶分歧问题的数值分析

构造了求解一类非退化分歧点及相关参数的扩充系统，给出了拟牛顿迭代法并证明了收敛性。     

Effect of constriction height on flow separation in a two-dimensional channel

We studied numerically the effect of the constriction height on viscous flow separation past a two-dimensional channel with locally symmetric constrictions. A numerically stable scheme in primitive variables (velocity and pressure) for the solution of two-dimensional incompressible time-dependent Navier–Stokes equations is employed using finite-difference approximation in staggered grid. The wall shear stresses at different heights of the constriction are computed and presented graphically. It is noticed that the maximum stress and the length of the recirculating region associated with two shear layers of the constriction increase with the increase of the area reduction of the constriction. The critical Reynolds number for symmetry breaking bifurcation for the 50%, 60% and 70% area reduction are obtained numerically. The flow field separates after the symmetry breaking bifurcation and the symmetry of the flow depends on the Reynolds number and the height of the constriction.     

Symmetry breaking of solutions of non-cooperative elliptic systems

In this article we study the symmetry breaking phenomenon of solutions of non-cooperative elliptic systems. We apply the degree for G$G$-invariant strongly indefinite functionals to obtain simultaneously a symmetry breaking and a global bifurcation phenomenon.     

Symmetry structure of the hyperbolic bifurcation without reflection of periodic orbits in the standard map

For the area preserving maps, the linearized tangent map determines the stability of the fixed point. When the trace of the tangent map is less than −2, the fixed point is inversion hyperbolic, thus the subsequent points of mapping alternate across the destabilized fixed point. That is to say, the fixed point undergoes periodic doubling bifurcation. While for the trace of the tangent map is larger than +2, the fixed point undergoes the hyperbolic bifurcation without reflection. Here, the processes of the hyperbolic bifurcation without reflection in the standard map have been examined in terms of the higher order symmetry in the momentum inversion. It is shown that the higher order symmetry lines approach asymptotically to the separatrix of the hyperbolic fixed point, and the existing symmetry lines cannot determine the structure of the periodic islands born after the hyperbolic bifurcation without reflection.     

Structural bifurcation of divergence-free vector fields near non-simple degenerate points with symmetry

In this study, topological features of an incompressible two-dimensional flow far from any boundaries is considered. A rigorous theory has been developed for degenerate streamline patterns and their bifurcation. The homotopy invariance of the index is used to simplify the differential equations of fluid flows which are parameter families of divergence-free vector fields. When the degenerate flow pattern is perturbed slightly, a structural bifurcation for flows with symmetry is obtained. We give possible flow structures near a bifurcation point. A flow pattern is found where a degenerate cusp point appears on the x-axis. Moreover, we also show that bifurcation of the flow structure near a non-simple degenerate critical point with double symmetry is generic away from boundaries. Finally, we give an application of the degenerate flow patterns emerging when index 0 and -2 in a double lid driven cavity and in two dimensional peristaltic flow.     

Computational bifurcation of periodic solutions in systems with symmetry

A bifurcation problem given by a -symmetric parameter-dependentsystem of autonomous ordinary differential equations is considered( a finite group). The main goal is to derive an efficient numericalmethod for the computation of symmetry breaking, period doublingand symmetry breaking period doubling bifurcations of periodicsolutions. For this the bifurcation problem is reformulatedin terms of Fourier coefficient vectors. This makes it possibleto make use of the spatial and temporal symmetries of periodicsolutions in order to reduce the effort for solving the problemeffectively. Finally, a Galerkin method based on Fourier expansionsis used for the numerical treatment and this method is illustratedby a numerical example.     

Rhombic Periodicity Lattice in Bifurcational Symmetry Breaking Problems

General theory of symmetry breaking problems in branching theory is presented in the works [1–3]. Such problems are invariant relative to Euclidean space motions group and their solution which is invariant relatively this group is the rest state or uniform linear motion. At the stability loss cell structure solutions arise, which are invariant relative to the group of the definite period shifts along the definite directions, passing mutually under discrete subgroup transformations that is defined by the symmetry of elementary cell of the periodicity. Thus the Euclidean space motions group is changed by the symmetry of a certain crystallographic group. In this article for the case of 4–dimensional degeneracy of the linearized Fredholm operator abstract bifurcational symmetry breaking problems both for stationary and Andronov‐Hopf bifurcation with planar rhombic periodicity lattice are considered. Applications to hydrodynamical problems are given.     

Subgroup structure of bifurcation equations with crystallographic group symmetry

For stationary and Andronov-Hopf bifurcation with symmetry of highest crystalline classes of basic syngonies of symmorphic crystallographic groups as semidirect products of translations on basic directions and point symmetry of lattice the subgroup structure of relevant bifurcation equations and bifurcating solutions dual to subgroup structure of such symmetry is investigated. Applications to phase transitions in physics are considered. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies

A numerical tool for the detection of degenerated symmetry breaking bifurcation points is presented. The degeneracies are classified and numerically processed on -D restrictions of the bifurcation equation. The test functions that characterise each of the equivalence classes are constructed by means of an equivariant numerical version of the Liapunov-Schmidt reduction. The classification supplies limited qualitative information concerning the imperfect bifurcation diagrams of the detected bifurcation points.

     

Geometric Invariant Theory in a Model-Independent Analysis of Spontaneous Symmetry and Supersymmetry Breaking

Functions which are covariant or invariant under the transformations of a reductive linear algebraic group can be advantageously expressed in terms of functions defined in the orbit space of the group, i.e. as functions of a finite set of basic invariant polynomials. This fact and the tools of geometric invariant theory can be exploited in many physical contexts where the study of covariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry and/or supersymmetry breaking in possibly supersymmetric quantum field theories of elementary particles, in the analysis of phase spaces and structural phase transitions in solid state physics (Landau's theory), in covariant bifurcation theory, in crystal field theory and in most areas of solid state theory where use is made of symmetry adapted functions. We shall present some elements of geometric invariant theory and illustrate some of the possible applications in the study of spontaneous symmetry and supersymmetry breaking.